Question: The equation of hyperbola $H$ is $\dfrac {(x+5)^{2}}{9}-\dfrac {(y-7)^{2}}{16} = 1$. What are the asymptotes?
We want to rewrite the equation in terms of $y$ , so start off by moving the $y$ terms to one side: $\dfrac {(y-7)^{2}}{16} = - 1 + \dfrac {(x+5)^{2}}{9}$ Multiply both sides of the equation by $16$ $(y-7)^{2} = { - 16 + \dfrac{ (x+5)^{2} \cdot 16 }{9}}$ Take the square root of both sides. $\sqrt{(y-7)^{2}} = \pm \sqrt { - 16 + \dfrac{ (x+5)^{2} \cdot 16 }{9}}$ $ y - 7 = \pm \sqrt { - 16 + \dfrac{ (x+5)^{2} \cdot 16 }{9}}$ As $x$ approaches positive or negative infinity, the constant term in the square root matters less and less, so we can just ignore it. $y - 7 \approx \pm \sqrt {\dfrac{ (x+5)^{2} \cdot 16 }{9}}$ $y - 7 \approx \pm \left(\dfrac{4 \cdot (x + 5)}{3}\right)$ Add $7$ to both sides and rewrite as an equality in terms of $y$ to get the equation of the asymptotes: $y = \pm \dfrac{4}{3}(x + 5)+ 7$